Is a marriage between a Muslim woman and a Shiite man considered a valid "Mut'ah" marriage in an American court, and what are the procedures for divorce in this case?

Praise be to Allah for guiding you to Islam. The fundamental principle of marriage is continuity and permanence. Mut'ah marriage (temporary marriage) was permissible at the beginning of Islam, then its prohibition was abrogated until the Day of Judgment. What occurred between you is an invalid marriage that must be annulled. Annulment can be by divorce or by other means. Regarding the dowry, the woman receives nothing if consummation has not occurred. However, after consummation, scholars have differed whether she is entitled to the stipulated dowry, the customary dowry (mahr al-mithl), or the lesser of the two. As for the waiting period ('iddah), it is obligatory after separation from this marriage to ensure the purity of the womb. Therefore, it is not permissible for you to remain in this invalid marriage. Rather, you must separate from it and observe the 'iddah, and then it becomes lawful for you to enter into a valid Islamic marriage after the 'iddah has ended.